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The Relevance of Artiﬁcial Intelligence for Human CognitionHelmar Gust and Kai–Uwe K¨uhnbergerInstitute of Cognitive ScienceAlbrechtstr.28University of Osnabr¨uck,Germany{hgust,kkuehnbe}@uos.deAbstractWe will discuss the question whether artiﬁcial intelli-gence can contribute to a better understanding of humancognition.We will introduce two examples in which AImodels provide explanations for certain cognitive abil-ities:The ﬁrst example examines aspects of analogicalreasoning and the second example discusses a possi-ble solution for learning ﬁrst-order logical theories byneural networks.We will argue that artiﬁcial intelli-gence can in fact contribute to a better understandingof human cognition.IntroductionQuite often artiﬁcial intelligence is considered as an engi-neering discipline,focusing on solutions for problems incomplex technical systems.For example,building a robotnavigation device in order to enable the robot to act in an un-known environment poses problems like the following ones:• How is it possible to detect obstacles by the sensory de-vices of the robot?• How is it possible to circumvent time-critical planningproblems of a planning system that is based on a timeconsuming deduction calculus?• Which problemsolving abilities are available to deal withoccurring unforeseen problems?• How is it possible to identify early enough dangerous ob-jects,surfaces,enemies etc.,in particular,if they are neverseen before by the robot?Although such problems do have certain similarities toclassical questions in cognitive science it is usually not as-sumed that solutions for the robot can be analogously trans-ferred to solutions for cognitive science.For example,a so-lution for a planning problem of a mobile robot does notnecessarily have any consequences for strategies to solveplanning problems in cognitive agents like humans.Quiteoften it is therefore claimed that engineering solutions fortechnical devices are not cognitively adequate.On the other hand,it is frequently assumed that cognitivescience and,in particular,the study of human cognition tryCopyrightc 2006,American Association for Artiﬁcial Intelli-gence (www.aaai.org).All rights reserved.to develop attempts for solutions of problems that are usu-ally considered as hard for artiﬁcial intelligence.Examplesare human abilities like adaptivity,creativity,productivity,motor coordination,perception,emotions,or goal genera-tion of autonomous agents.It seems to be the case that theseaspects of human cognition do not have simple solutions thatcan be straightforwardly implemented in a machine.There-fore,human cognition is often considered as a reservoir fornewchallenges in artiﬁcial intelligence.In this paper,we will discuss the question:Can artiﬁ-cial intelligence contribute to our understanding of humancognition?We will argue for the existence of such a con-tribution (contrary to the discussion above).Our argumentsare based on own results in two domains of AI research:ﬁrst,analogical reasoning and second,the learning of logicalﬁrst-order theories by neural networks.We claimthat appro-priate solutions in artiﬁcial intelligence can provide explana-tions in cognitive science by using well-established formalmethods,the rigorous speciﬁcation of the problem,and thepractical realization in a computer program.More preciselyby modeling analogical reasoning with formal tools we willbe able to get an idea how creativity and productivity of hu-man cognition as well as efﬁcient learning without large datasets is possible.Furthermore,learning ﬁrst-order theories byneural networks can be used to explain why human cogni-tion is often model-based and less time-consuming than aformal deduction.Therefore,artiﬁcial intelligence can con-tribute to a better understanding of human cognition.The paper will have the following structure:First,we willdiscuss an account for modeling analogical reasoning andwe will sketch the consequences for cognitive science.Sec-ond,we will roughly discuss the impact of a solution forlearning logical inferences by neural networks.We will givean explanation for some empirical ﬁndings.Finally we willsummarize the discussion.Example 1:Analogical ReasoningThe Analogy between Water and HeatIt is quite undisputed that analogical reasoning is an impor-tant aspect of human cognition.Although there has been astrong endeavor during the last twenty-ﬁve years to developa theory of analogies and,in particular,a theory of analog-ical learning,no generally accepted solution has been pro-Figure 1:The diagrammatic representation of the heat-ﬂowanalogy.posed yet.Connected with the problem of analogical rea-soning is the problem of interpreting metaphorical expres-sions (Gentner et al.2001).Similarly to analogies,there isno convincing automatic procedure that computes the mean-ing of metaphorical expressions in a broad variety of do-mains either.Recently published,the monograph (Gentner,Holyoak,and Kokinov 2001) can be seen as a summary ofimportant approaches towards a modeling of analogies.Figure 1 represents the analogy between a water-ﬂowsys-tem,where water is ﬂowing fromthe beaker to the vial,anda heat-ﬂow system,where heat is ﬂowing from the warmcoffee to a berylliumcube.The analogy consists of the asso-ciation of water-ﬂowon the source side and heat-ﬂowon thetarget side.Although this seems to be a rather simple anal-ogy,a non-trivial property of this association must be mod-eled:The concept heat is a theoretical termand not anythingthat can be measured directly by a physicist.Therefore theestablishment of an analogical relation between the water-ﬂow system and the heat-ﬂow system must productivelygenerate an additional concept heat ﬂowing fromwarmcof-fee to a cold berylliumcube.This leads to an analogy wherethe measureable heights of the water levels in the beaker andthe vial correspond to the temperature of the warm coffeeand the temperature of the berylliumcube,respectively.HDTP – A Theory Computing AnalogiesHeuristic-Driven Theory Projection (HDTP) is a formallysound theory for computing analogical relations betweena source domain and a target domain.HDTP computesanalogical relations not only by associating concepts,rela-tions,and objects,but also complex rules and facts betweenthe target and the source domain.In (Gust,K¨uhnberger,and Schmid 2005a) the syntactic,semantic,and algorith-mic properties of HDTP are speciﬁed.Unlike to well-knownaccounts for modeling analogies like the structure-mappingengine (Falkenhainer,Forbus,and Gentner 1989) or Copy-cat (Hofstadter 1995),HDTP produces abstract descriptionsof the underlying domains,is heuristic-driven,i.e.allows toinclude various types of background knowledge,and has amodel theoretic semantics induced by an algorithm.Syntactically,HDTP is deﬁned on the basis of a many-sorted ﬁrst-order language.First-order logic is used in or-der to guarantee the necessary expressive power of the ac-count.An important assumption is that analogical reasoningTable 1:A simpliﬁed description of the algorithm HDTP-A omitting formal details.A precise speciﬁcation of thisalgorithm can be found in (Gust,K¨uhnberger,and Schmid2005a).Input:A theory ThSof the source domain and a theoryThTof the target domain represented in a many-sorted predicate logic language.Output:A generalized theory ThGsuch that the inputtheories ThSand ThTcan be reestablished bysubstitutions.Selection and generalization of fact and rules.Select an axiomfromthe target domain(according to a heuristics h).Select an axiomfromthe source domain andconstruct a generalization (together withcorresponding substitutions).Optimize the generalization w.r.t.a given heuristics h

.Update the generalized theory w.r.t.the result ofthis process.Transfer (project) facts of the source domain to the targetdomain provided they are not generalized yet.Test (using an oracle) whether the transfer isconsistent with the target domain.crucially contains a generalization (or abstraction) process.In other words,the identiﬁcation of common properties orrelations is represented by a generalization of the input ofsource and target.Formally this can be modeled by an ex-tension of the so-called theory of anti-uniﬁcation (Plotkin1970),a mathematically sound account describing the pos-sibility of generalizing terms of a given language using sub-stitutions.More precisely,an anti-uniﬁcation of two termst1and t2can be interpreted as ﬁnding a generalized termt (or structural description t) of t1and t2which may con-tain variables,together with two substitutions Θ1and Θ2ofvariables,such that tΘ1= t1and tΘ2= t2.Because thereare usually many possible generalizations,anti-uniﬁcationtries to ﬁnd the most speciﬁc one.An example should makethis idea clear.Assume two terms t1= f(x,b,c) andt2= f(a,y,c) are given.Generalizations are,for exam-ple,the terms t = f(x,y,c) and t

= f(x,y,z) togetherwith their corresponding substitutions.1But t is more spe-ciﬁc than t

,because the substitution Θ substituting z by ccan be applied to t

.This application results in:t

Θ = t.Most speciﬁc generalizations of two terms are commonlycalled anti-instances.Given two input theories ThSand ThTfor source andtarget domains respectively,the algorithm HDTP-A com-putes anti-instances together with a generalized theory ThG.Table 1 makes the algorithm more precise:First,an axiomfromthe target domain is selected guided by an appropriateheuristics h,for example,measuring the syntactic complex-ity of the axiom.Then an axiom of the source domain issearched in order to construct a generalization together with1As usual we assume that a,b,c,...denote constants andx,y,z,...denote variables.Table 2:Examples of corresponding concepts in the source and the target domains of the heat-ﬂowanalogy.SourceTargetA(1) connected(beaker,vial,pipe)connected(coffe in cup,bcube,bar)connected (A,B,C)(2) liquid(water)liquid(coffee)liquid(D)(3) height(water in beaker,t1) >height(water in vial,t2)temp(coffe in cup,t1) >temp(bcube,t1)T(A,t1) >T(B,t1)(4) height(water in beaker,t1) >height(water in beaker,t2)temp(coffee in cup,t1) >temp(coffe in cup,t2)T(A,t1) >T(A,t2)(5) height(water in vial,t2) >height(water in vial,t1)temp(bcube,t2) >temp(bcube,t1)T(B,t2) >T(B,t1)substitutions.The generalization is optimized using anotherheuristics h

,for example,the length of the necessary sub-stitutions.Finally axioms from the source domain are pro-jected to the target domain.Then the transferred axioms aretested for consistency with the target domain using an ora-cle.Applying this theory to our example depicted in Figure 1yields the intuitively correct result.Table 2 depicts some ofthe crucial associations that are important for establishingthe analogy.We summarize the corresponding substitutionsΘ1and Θ2in the following list:A −→beaker/coffee in cupB −→vial/bcubeC −→pipe/barD −→water/coffeeT −→λx,t:height(water in x,t)/temperatureThe example – although seemingly simple – has a rela-tively complicated aspect:The systemassociates an abstractproperty λx,t:height(water in x,t) with temperature.Theconcept heat must be introduced as a counterpart of waterin the target domain by projecting the structure of the λ-term above to the target domain by the following equation:temperature(x,t) = height(heat in x,t)HDTP was applied to a variety of domains,for example,naive physics (Schmid et al.2003) and metaphors (Gust,K¨uhnberger,and Schmid 2005b).The algorithmHDTP-Aisimplemented in SWI-Prolog.The core programis availableonline (Gust,K¨uhnberger,and Schmid 2003).Explanations for Cognitive ScienceWe would like to argue for the claim that the sketched pro-ductive solutions of analogical reasoning problems can havean impact to the understanding of human cognition.Theﬁrst argument is that HDTP is a theory and speciﬁes analog-ical reasoning on a syntactic,a semantic,and an algorithmiclevel.This is quite often different in frameworks developedfroma cognitive science perspective.Usually those accountsgive precise descriptions of psychological experiments,of-ten they try to ﬁnd psychological generalizations,but regu-larly they lack a formally speciﬁed explanation why someempirical data can be measured.The advantage of an AIsolution to analogies is that a ﬁne-grained analysis of anal-ogy making can be achieved due to the formally speciﬁedlogical basis and the algorithmic speciﬁcation.This enablesus to specify precisely which assumptions must be made inorder to be able to establish analogical relations.Analogical reasoning shows an important feature that dis-tinguishes this type of inferences from other types of rea-soning,like inductive learning,case-based reasoning,orexemplar-based learning:All these latter forms of learn-ing are based on a rather large number of training instanceswhich are usually barely structured.Learning is possible,because many instances are available.Therefore generaliza-tions of existing data can primarily be computed due to alarge number of examples,whereas given domain theoriesplay usually a less important role.In contrast to these typesof learning,analogical learning is based on a rather smallnumber of examples:In many cases only one (rich) concep-tualization of the source domain and a rather coarse concep-tualization of the target domain is available.But on the otherhand analogies are based on sufﬁcient background knowl-edge.A cognitive science explanation for analogical infer-ences must take this into account.It is not sufﬁcient to applystandard learning algorithms to explain analogical learning,but accounts need to be used that explain precisely why thebackground knowledge is sufﬁcient in one application butinsufﬁcient in another.Furthermore,accounts are neededthat can explain whether a particular analogical relation canbe established without taking into account a spelled-out the-ory,or whether such a theory is in fact necessary.Preciselythis can be achieved by applying HDTP.Because the discovery of a sound analogical relation pro-vides immediately a new conceptualization of the target do-main,this may be a hint for the explanation of sudden in-sights.Notice that such insights could have a certain con-nection to the Gestalt laws.Such Gestalt laws can be inter-preted as the concurrency of different analogical relations.Therefore analogical reasoning can be extended to furtherhigher cognitive abilities.We summarize why the modeling of analogies usingHDTP contributes to the understanding of human cognition:• HDTP is a theory,not a description of empirical data,ex-plaining productive capabilities of human cognition.• HDTP provides a ﬁne-grained analysis of analogicaltransfers on a syntactic,semantic,and algorithmic level.• HDTP provides an explanation why analogical learning ispossible without a large number of examples.• An extension to other cognitive abilities seems to bepromising.Example 2:Symbols and Neural NetworksThe ProblemThe gap between symbolic and subsymbolic models of hu-man cognition is usually considered as a hard problem.Onthe symbolic level recursion principles ensure that the for-malisms are productive and allow a very compact represen-tation:Due to the compositionality principle it is possibleto compute the meaning of a complex (logical) expressionusing the meaning of the embedded subexpressions.Onthe other hand,it is assumed that neural networks are non-compositional on a principal basis making it difﬁcult to rep-resent complex data structures like lists,trees,tables,for-mulas etc.Two aspects can be distinguished:The represen-tation problem (Barnden 1989) and the inference problem(Shastri and Ajjanagadde 1990).The ﬁrst problem statesthat,if at all,complex data structures can only be used im-plicitly and the representation of structured objects is a non-trivial challenge for neural networks.The second problemtries to model inferences of logical systems with neural ac-counts.A certain endeavor has been invested to solve the repre-sentation problem as well as the inference problem.It iswell-known that classical logical connectives like conjunc-tion,disjunction,or negation can be represented by neuralnetworks.Furthermore it is known that every Boolean func-tion can be learned by a neural network (Steinbach and Ko-hut 2002).Although it is therefore possible to representpropositional logic with neural networks,this is not true forﬁrst-order logic (FOL).The corresponding problem,usuallycalled the variable-binding problem,is caused by the usageof quantiﬁers ∀ and ∃,which may bind variables that occurat different positions in one and the same formula.There area number of attempts to solve the problem of representinglogical formulas with neural networks:Examples are signpropagation (Lange and Dyer 1989),dynamic localist repre-sentations (Barnden 1989),or tensor product representations(Smolensky 1990).Unfortunately these accounts have cer-tain non-trivial side-effects.Whereas sign propagation anddynamic localist representations lack the ability of learning,tensor product representations result in an exponentially in-creasing number of elements to represent variable bindings,just to mention some of the problems.With respect to the inference problem of connectionistnetworks the number of proposed solutions is rather smalland relatively new.An attempt is (Hitzler,H¨olldobler,andSeda 2004) in which a logical deduction operator is approx-imated by a neural network.In (D’Avila Garcez,Broda,and Gabbay 2002),tractable fragments of predicate logic arelearned by connectionist networks.Closing the Gap between Symbolic andSubsymbolic RepresentationsIn (Gust and K¨uhnberger 2004) and (Gust and K¨uhnberger2005) a framework was developed that enables neural net-works to learn logical ﬁrst-order theories.The idea is rathersimple:Because interpretation functions of FOL cannot belearned directly by neural networks (due to their heteroge-neous structure and the variable-binding problem) logicalformulas are translated into a homogeneous variable-freerepresentation.The underlying structure for this represen-tation is a topos (Goldblatt 1984),a category theoretic struc-ture that can be interpreted as a model of FOL (Gust 2000).In a topos,logical expressions correspond simply to con-structions of arrows given other arrows.Therefore everyconstruction can be reduced to one single operation,namelyLOGICInput:A set oflogicalformulasgiven ina logicallanguage⇒✻Is done by hand butcould easily be doneby a programTOPOSTheinput istranslatedinto a setof objectsandarrows⇒✻PROLOGpro-gramf ◦g = hEquationsin normalform areidenti-fyingarrows inthe topos⇒✻The equations generated by thePROLOG program are used asinput for the neural networkNNsLearning:achievedby min-imizingdistancesbetweenarrowsFigure 2:The general architecture of an account transferringlogical theories into a variable-free representation and feed-ing a neural network with equations of the formf ◦ h = g.the concatenation of arrows,i.e.the concatenation of set-theoretic functions (in the easiest case that the topos is iso-morphic to the category SETof sets and set theoretic func-tions).In a topos,not every arrow corresponds directly toa symbol (or a complex string of symbols).Similarly thereare symbols that have no direct representation in a topos:For example,variables do not occur in a topos but are hid-den or indirectly represented.Another example of symbolsthat have no simple representation in a topos are quantiﬁers.Figure 2 depicts the general architecture of the system.Given a representation of a ﬁrst-order logical formula in atopos,a Prolog program generates equations f ◦ g = hof arrows in normal form that can be fed to a neural net-work.The equations are determined by constructions thatexist in a topos.Examples are products,coproducts,or pull-backs.2The network is trained using these equations anda simple backpropagation algorithm.Due to the fact that atopos codes implicitly symbolic logic,we call the represen-tation of logic in a topos the semisymbolic level.In a topos,an arrow connects a domain with a codomain.In the neuralrepresentations,all these entities (domains,codomains,andarrows) are represented as points in a n-dimensional vectorspace.The structure of the network is depicted in Figure 3.Inorder to enable the system to learn logical inferences,somebasic arrows have static (ﬁxed) representations.These rep-resentations correspond directly to truth values.• The truth value true:(1.0,1.0,1.0,1.0,1.0)• The truth value false:(0.0,0.0,0.0,0.0,0.0)Notice that the truth value true and the truth value falseare maximally distinct.First results of learning FOL by thisapproach are promising (Gust and K¨uhnberger 2005).Both,the concatenationoperationand the representations of the ar-rows together with their domains and codomains are learnedby the network.Furthermore the network does not only learna certain input theory but rather a model of the input theory,2Simply examples in set theory for product constructions areCartesian products.Coproducts correspond to disjoint unions ofsets.Pullbacks are generalized products.first layer: 5*n hidden layer: 2*n output layer: ndom1a1cod1=dom2a2cod2a2 ◦ a11Figure 3:The structure of the neural network that learnscomposition of ﬁrst-order formulas.i.e.the input together with the closure of the theory under adeduction calculus.The translation of ﬁrst-order formulas into training data ofa neural network allows,in principal,to represent models ofsymbolic theories in artiﬁcial intelligence and cognitive sci-ence (that are based on FOL) with neural networks.3In otherwords the account provides a recipe – and not just a generalstatement of the possibility – of how to learn models of the-ories based on FOL with neural networks.The sketched ac-count tries to combine the advantages of connectionist net-works and logical systems:Instead of representing symbolslike constants or predicates using single neurons,the rep-resentation is rather distributed,realizing the very idea ofdistributed computations in neural networks.Furthermorethe neural network can be trained quite efﬁciently to learn amodel without any hardcoded devices.The result is a dis-tributed representation of a symbolic system.4Explaining Inferences as the Learning of ModelsA logical theory consists of axioms specifying facts andrules about a certain domain together with a calculus deter-mining the “correct” inferences that can be drawn fromtheseaxioms.From a computational point of view this quite of-ten generates problems,because inferences can be rather re-source consuming.Modeling logical inferences with neuralnetworks as sketched in the subsection above allows a veryefﬁcient way of drawing inferences,simply because the in-terpretation of possible queries is “just there”,namely im-plicitly coded by the distribution of the weights of the net-work.The account explains why time-critical deductionscan be performed by humans using models instead of cal-culi.It is important to emphasize that the neural networkdoes not only learn the input,but a whole model makingthe input true.In a certain sense these models are overde-termined,i.e.they assign truth values to every query,eventhough the theory does not determine a truth value.Never-theless the models are consistent with the theory.This dis-3Notice that a large part of theories in artiﬁcial intelligence areformulated with tools taken from logic and are mostly based onFOL or subsystems of FOL.4In a certain sense the presented account is an extreme caseof a distributed representation,opposing the other extreme case ofa purely symbolic representation.Human cognition is probablyneither of the two extreme cases.tinguishes the trained neural network from a symbolic the-orem prover.Whereas the theorem prover just deduces thetheorems of the theory consistent with the underlying logic,the neural network assigns values to every query.There is empirical evidence from the famous Wasonselection-task (and the various versions of this task) that hu-man behavior is (in our terminology) rather model-basedthan theory-based,i.e.human behavior can be deduc-tive without having an inference mechanism(Johnson-Laird1983).In other words,humans do not performdeductions ifthey reason logically,but rather apply a model of the corre-sponding situation.We can give an explanation of this phe-nomenon using the presented neural network account:Hu-mans act mostly according to a model they learned (about,for example,a situation,a scene,or a state of affairs) and notaccording to a theory plus an inference mechanism.There is a certain tendency of our learned models towardsa closed-world assumption.Consider the following rules:All humans are mortal.All mortal beings ascend to heaven.All beings in heaven are angels.If we know that Socrates is human we would like to de-duce that Socrates is an angel.But if we just know that therobot is not mortal,we would rather like to deduce that therobot is not an angel.The models learned by the neural net-work provide hints for an explanation of these empirical ob-servations:The property of the robot to be non-human prop-agates to the property of the robot to be non-angel.Thisprovides evidence for an equivalence between The robot ishuman and The robot is an angel in certain types of under-determined situations.A difﬁcult problem for cognitive science and symbol-based robotics are modelings of time constraints.On theone hand,it is possible for humans to be quite successfulin a hostile environment in which time-critical situations oc-cur and rapid responses and actions involving some kind ofplanning are necessary.On the other hand,symbol-basedmachines often have signiﬁcant problems in solving suchtasks.A natural explanation is that humans do not deduceanything,but rather apply an appropriate model in certaincircumstances.Again this type of explanation can be mod-eled by the sketched connectionist approach.All knowledgeabout a state of affairs is just there,namely implicitly codedin the weights of the network.Clearly,the correspondingmodel can be wrong or imprecise,but a reaction in time-critical situations is always possible.Although the gap between symbolic and subsymbolic ap-proaches in cognitive science and AI is obvious,there is stillno generally accepted solution for this problem.In partic-ular,in order to understand human cognition the questionis often raised of how an explanation for the emergence ofconceptual knowledge from subsymbolic sensory data andthe emergence of subsymbolic motor behavior from con-ceptual knowledge are possible at all.To put the task intothe symbol-neural distinction (without discussing the differ-ences between the two formulations):how can rules be re-trieved fromtrained neural networks and how can symbolicknowledge (including complex data structures) be learnedby neural networks.Clearly we do not claim to solve thisproblem,but at least our approach shows how one direction– namely the learning of logical ﬁrst-order theories by neuralnetworks – can uniformly be solved.In this approach twomajor principles are realized:ﬁrst,the network can learn,and second,the topology of the network does not need to bechanged in order to learn new input.We do not know anyother approach that realizes these two principles.Again we summarize the arguments why an AI solutionfor logical inferences using neural networks can contributeto the understanding of human cognition:• The presented account explains why logical inferences areoften based on models or situations not on logical deduc-tions.• It is possible to explain,why complex inferences can berealized by humans but are rather time-consuming to re-alize for deduction calculi.• Last but not least,we can give hints how neural networks– usually considered as inappropriate for the deduction oflogical facts – can be used to performlogical inferences.ConclusionsIn this paper,we discussed two AI models that provide solu-tions for certain aspects of higher cognitive abilities.Thesemodels were used to argue for the claim that artiﬁcial intel-ligence can contribute to a better understanding of humancognition.In particular,we argued that the computation ofanalogies using HDTP can explain the creativity of analogi-cal inferences in a mathematically sound framework withoutreference to a large number of examples.Furthermore we ar-gued 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